Augmenting the great icosidodecahedron

The great icosidodecahedron, vertex figure (5/2,3,5/2,3)

Left: Each triangular face of the great icosidodecahedron can be augmented with a triangular pyramid or tetrahedron.  Each 3 in the vertex figure is replaced by (3,3)   The original vertex figure becomes (5/2,3,3,5/2,3,3) with new vertices of (3,3,3).  All vertices are on the exterior of the polyhedron.

Right: The above figure is shown with the tetrahedra shown in framework style. A framework of this model is linked here.

Left: Another augmentation of the great icosidodecahedron can be obtained by excavating each pentagrammic face with a pentagrammic pyramid completing a cycle around the axis and each triangular face with a triangular pyramid or tetrahedron which again completes a cycle around the axis.  Each 5/2 and each 3 in the vertex figure is replaced by (3,3)   The original vertex figure becomes (3,3,3,3,3,3,3,3)/3 or (38)/3 with new vertices of (3,3,3,3,3)/2 and (3,3,3) respectively.  The faces of the pentagrammic pyramids are shown in yellow and the tetrahedral faces in blue.  All vertices lie on the convex hull.

Centre: A further augmentation of the great icosidodecahedron can be obtained by excavating each pentagrammic face with a pentagrammic antiprism which completing a cycle around the axis and each triangular face with a triangular pyramid or tetrahedron which again completes a cycle around the axis.  Each 5/2 in the vertex figure is replaced by a (3,3,3) and each 3 in the vertex figure is replaced by (3,3)   The original vertex figure becomes (3,3,3,3,3,3,3,3,3,3)/3 or (310)/3 with new vertices of (5/2,3,3,3) and (3,3,3) respectively.  The triangular faces of the pentagrammic antiprisms are shown in yellow and the tetrahedral faces in blue.  All vertices are on the exterior of the polyhedron.

Right: A further augmentation of the great icosidodecahedron can be obtained as follows:  Excavate one pentagrammic cap of a crossed pentagrammic antiprism with a pentagrammic pyramid.  The resulting polyhedron is equivalent to a diminished great icosahedron.  This polyhedron is not locally convex but the act of joining it to the great icosidodecahedron removes the retrograde polygon .  Excavate each of the pentagrammic faces of the great icosidodecahedron with a diminished great icosahedron.  The 5/2 in the vertex figure is replaced by a (3,3,3)/2   The original vertex figures become (3,3,3,3,3,3,3,3)/3 - or (38)/3 with new vertices from the diminished great icosahedron of (3,3,3,3,3)/2 - or (35)/2.  The original triangular faces of the stellatruncated cube are shown in blue, those of the anti-prisms in yellow and the pyramids red.  All vertices are on the exterior of the polyhedron.